The assumption that MFCQ holds at every \(w\in C\)  implies that \(C\)  is Clarke regular (i.e. if \((w_k, u_k) \rightarrow (w,u) \in C \times \mathbb {R}^d\)  with \((w_k, u_k) \in C \times N_{C}(w_k)\) , then \(u \in N_{C}(w)\) ). As shown in (Sections 5 and 6 [1]}), this implies that for almost every \(t \ge 0\)  and every \(v \in N_{C}(\mathsf {y}(t))\) , \(\dot{\mathsf {y}}(t)^\top v = 0\) . Therefore, for almost every \(t \ge 0\) ,
\(\frac{\mathrm {d}}{\mathrm {d}t}\ell (\mathsf {y}(t)) &= \nabla \ell (\mathsf {y}(t))^\top \dot{\mathsf {y}}(t) \\&\in - \left\Vert \dot{\mathsf {y}}(t) \right\Vert ^2 - \dot{\mathsf {y}}(t)^\top \mathsf {v}(t) = - \left\Vert \dot{\mathsf {y}}(t) \right\Vert ^2 \, ,\) 
